Chapter 16.2, Problem 48E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

The base of a circular fence with radius 10 m is given by *x =* 10 cos *t*, *y* = 10 sin *t.* The height of the fence at position (*x*, *y*) is given by the function *h*(*x, y*) = 4 + 0.01(*x*^{2} − *y*^{2}), so the height varies from 3 m to 5 m. Suppose that 1 L of paint covers 100 m^{2}. Sketch the fence and determine how much paint you will need if you paint both sides of the fence.

To determine

**To sketch:** The circular fence and paint required to pain both sides of the fence.

Explanation

**Given data:**

The radius of the circular fence is 10 m.

The parameters are given as follows.

The height of the circular fence is varied from 3 m to 5 m and the function of height is given as follows.

1 L of paint covers the area of

**Formula used:**

Write the expression to find area of the fence.

Here,

Write the expression to find

Write the expression to find paint required.

Consider the base of the circular fence lies in the *xy*-plane.

Consider the center of the circular fence lies in the origin with the height

From equation (1) and the given data (height varies from 3 m to 5 m), notice that the fence has highest height of 5m when the parameter

From equation (1), the height of the fence is 4 m when the parameter

Consider the parameter *t* as *u*. and the write the parameters *x* and *y* as follows.

As the
*z* as follows.

Substitute

From the parameters, write the parametric equations of the fence as follows.

Use the parametric equations and sketch the fence as shown in Figure 1.

Thus, the fence is sketched for the given data.

**Calculation of**
**:**

Substitute

**Calculation of area of fence:**

Substitute

Sect-16.1 P-1ESect-16.1 P-2ESect-16.1 P-3ESect-16.1 P-4ESect-16.1 P-5ESect-16.1 P-6ESect-16.1 P-7ESect-16.1 P-8ESect-16.1 P-9ESect-16.1 P-10E

Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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